Revision as of 18:26, 17 August 2012

Hask refers to a category with types as objects and functions between them as morphisms. However, its use is ambiguous. Sometimes it refers to Haskell (actual Hask), and sometimes it refers to some subset of Haskell where no values are bottom and all functions terminate (platonic Hask). The reason for this is that platonic Hask has lots of nice properties that actual Hask does not, and is thus easier to reason in. There is a faithful functor from platonic Hask to actual Hask allowing programmers to think in the former to write code in the latter.

1 Hask

Actual Hask does not have sums, products, or an initial object, and

()

is not a terminal object. The Monad identities fail for almost all instances of the Monad class.

Why Hask isn't as nice as you'd thought.

Initial Object

Terminal Object

Sum

Product

Definition

There is a unique function

u :: Empty -> r

There is a unique function

u :: r ->()

For any functions

f :: a -> r

g :: b -> r

there is a unique function

u ::Either a b -> r

such that:

u . Left = f

u . Right = g

For any functions

f :: r -> a

g :: r -> b

there is a unique function

u :: r ->(a,b)

such that:

fst. u = f

snd. u = g

Platonic candidate

u1 r =case r of{}

u1 _=()

u1 (Left a)= f a

u1 (Right b)= g b

u1 r =(f r,g r)

Example failure condition

r ~()

r ~()

r ~()

f _=()

g _=()

r ~()

f _=undefined

g _=undefined

Alternative u

u2 _=()

u2 _=undefined

u2 _=()

u2 _=undefined

Difference

u1 undefined=undefined

u2 undefined=()

u1 _=()

u2 _=undefined

u1 undefined=undefined

u2 undefined=()

u1 _=(undefined,undefined)

u2 _=undefined

Result

FAIL

FAIL

FAIL

FAIL

2 Platonic Hask

Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottoms. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products. Instances of Functor and Monad really are endofunctors and monads.